calculate the derivative of the following function.\n$y = 2\\log_2(x^2 - 2)$\n$\frac{d}{dx}2\\log_2(x^2…

calculate the derivative of the following function.\n$y = 2\\log_2(x^2 - 2)$\n$\frac{d}{dx}2\\log_2(x^2 - 2)=\\square$

calculate the derivative of the following function.\n$y = 2\\log_2(x^2 - 2)$\n$\frac{d}{dx}2\\log_2(x^2 - 2)=\\square$

Answer

Explanation:

Step1: Use the logarithmic derivative formula

The derivative of (\log_a u) is (\frac{1}{u\ln a}\cdot u'). For (y = 2\log_2(x^2 - 2)), let (u=x^2 - 2). Then (y'=2\cdot\frac{1}{(x^2 - 2)\ln 2}\cdot(x^2 - 2)').

Step2: Differentiate (u=x^2 - 2)

Using the power rule ((x^n)'=nx^{n - 1}), ((x^2 - 2)'=2x).

Step3: Simplify the expression

Substitute ((x^2 - 2)' = 2x) into the formula: (y'=2\cdot\frac{1}{(x^2 - 2)\ln 2}\cdot2x=\frac{4x}{(x^2 - 2)\ln 2}).

Answer:

(\frac{4x}{(x^2 - 2)\ln 2})